Homology and cell structure of nilpotent spaces

Author:
Robert H. Lewis

Journal:
Trans. Amer. Math. Soc. **290** (1985), 747-760

MSC:
Primary 55P99; Secondary 20C07, 57M99

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792825-4

MathSciNet review:
792825

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let and denote finitely dominated nilpotent complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex.

We investigate the set of possible cofibers of maps , and find a severe restriction. When it is met and the fundamental group is finite, can be constructed from by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes.

We show that the Euler characteristic of must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes.

Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.

**[1]**K. S. Brown,*Euler characteristics of discrete groups and*-*spaces*, Invent. Math.**27**(1974), 229-264. MR**0385007 (52:5877)****[2]**K. S. Brown and P. K. Kahn,*Homotopy dimension and simple cohomological dimension of spaces*, Comment. Math. Helv.**52**(1977), 111-127. MR**0438336 (55:11251)****[3]**G. Carlsson,*A counterexample to a conjecture of Steenrod*, Invent. Math.**64**(1981), 171-174. MR**621775 (82j:57036)****[4]**W. Dwyer,*Vanishing homology over nilpotent groups*, Proc. Amer. Math. Soc.**49**(1975), 259-261. MR**0374242 (51:10442)****[5]**K. Ehrlich,*The obstruction to the finiteness of the total space of a fibration*, Michigan Math. J.**28**(1981), 19-38. MR**600412 (82i:55014)****[6]**K. Hoechsmann, P. Roquette and H. Zassenhaus,*A cohomological characterization of finite nilpotent groups*, Arch. Math. (Basel)**19**(1968), 225-244. MR**0227278 (37:2863)****[7]**P. J. Hilton,*Homotopy theory and duality*, Gordon & Breach, New York, 1966.**[8]**R. H. Lewis,*Equivariant cofibrations and nilpotency*, Trans. Amer. Math. Soc.**267**(1981), 139-155. MR**621979 (82j:55009)****[9]**-, Ph. D. dissertation, Cornell University, 1977.**[10]**G. Mislin,*Wall's obstruction for nilpotent spaces*, Topology**14**(1975), 311-317. MR**0394647 (52:15448)****[11]**-,*Finitely dominated nilpotent spaces*, Ann. of Math. (2)**103**(1976), 547-556. MR**0415607 (54:3690)****[12]**-,*Groups with cyclic Sylow subgroups...*, Comment. Math. Helv.**52**(1977), 373-391. MR**0448344 (56:6651)****[13]**-,*The geometric realization of Wall obstructions...*, Math. Proc. Cambridge Philos. Soc.**87**(1980), 199-206. MR**553576 (81f:55011)****[14]**G. Mislin and K. Varadarajan,*The finiteness obstructions for nilpotent spaces lie in*, Invent. Math.**53**(1979), 185-191. MR**560413 (81e:55011)****[15]**D. J. S. Robinson,*The vanishing of certain homology and cohomology groups*, J. Pure Appl. Algebra**7**(1976), 145-167. MR**0404478 (53:8280)****[16]**D. S. Rim,*Modules over finite groups*, Ann. of Math. (2)**69**(1959), 700-712. MR**0104721 (21:3474)****[17]**J. Stallings,*Homology and central series of groups*, J. Algebra**2**(1965), 170-181. MR**0175956 (31:232)****[18]**R. Swan,*Periodic resolutions for finite groups*, Ann. of Math. (2)**72**(1960), 267-291. MR**0124895 (23:A2205)****[19]**-,*Induced representations and projective modules*, Ann. of Math. (2)**71**(1960), 552-578. MR**0138688 (25:2131)****[20]**-,*The Grothendieck ring of a finite group*, Topology**2**(1963), 85-110. MR**0153722 (27:3683)****[21]**C. T. C. Wall,*Finiteness conditions for*-*complexes*, Ann. of Math. (2)**81**(1965), 56-69. MR**0171284 (30:1515)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55P99,
20C07,
57M99

Retrieve articles in all journals with MSC: 55P99, 20C07, 57M99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792825-4

Article copyright:
© Copyright 1985
American Mathematical Society